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1.2PureFoundation

Expanding Brackets

Expanding means multiplying out brackets so an expression has no brackets left. It is the reverse of factorising and shows up everywhere — from solving equations to differentiating. The golden rule never changes: every term inside must be multiplied by every term outside.

30 min Algebraic Expressions
Edexcel AS Level Maths: 1.2 Expanding Brackets (Double/Triple)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Expand a single bracket by multiplying through
  • Expand two brackets using a reliable method (FOIL / grid)
  • Expand and simplify expressions with three brackets
  • Collect like terms accurately after expanding
  • Handle negative signs and coefficients without slips
1

Expanding a single bracket

To expand a single bracket, multiply the term outside by each\textbf{each} term inside. Watch the signs carefully — a negative outside the bracket flips the sign of everything inside.

a(b+c)=ab+aca(b + c) = ab + ac
The multiplier aa is distributed across every term in the bracket.
Example 1Expand 3x(2x−5)3x(2x - 5)
1Multiply 3x3x by 2x2x: 6x26x^{2}.
2Multiply 3x3x by −5-5: −15x-15x.
Answer6x2−15x6x^{2} - 15x
Example 2Expand −2(4−3x)-2(4 - 3x)
1Multiply −2-2 by 44: −8-8.
2Multiply −2-2 by −3x-3x: +6x+6x (negative × negative = positive).
Answer−8+6x-8 + 6x

Tip — A negative sign outside the bracket is a multiplier of −1. Treat it like any other factor and the signs take care of themselves.

2

Expanding two brackets

For two brackets, every term in the first must multiply every term in the second — four products in total. A common memory aid is FOIL\textbf{FOIL}: First, Outer, Inner, Last. A grid (box) method does exactly the same job and is harder to rush.

(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^{2} + (a + b)x + ab
The middle term is the sum a+ba+b; the constant is the product abab.
Example 1Expand (x+4)(x+7)(x + 4)(x + 7)
1First: x×x=x2x \times x = x^{2}.
2Outer + Inner: 7x+4x=11x7x + 4x = 11x.
3Last: 4×7=284 \times 7 = 28.
Answerx2+11x+28x^{2} + 11x + 28
Example 2Expand (2x−3)(x+5)(2x - 3)(x + 5)
1First: 2x×x=2x22x \times x = 2x^{2}.
2Outer: 2x×5=10x2x \times 5 = 10x. Inner: −3×x=−3x-3 \times x = -3x.
3Combine the middle terms: 10x−3x=7x10x - 3x = 7x.
4Last: −3×5=−15-3 \times 5 = -15.
Answer2x2+7x−152x^{2} + 7x - 15

Tip — Always combine the two “middle” terms. Leaving them separate is the single most common expanding error.

3

A useful special case

Squaring a bracket is just expanding it by itself. The shortcut (x+a)2=x2+2ax+a2(x+a)^2 = x^2 + 2ax + a^2 saves time, but only if you remember the middle 2ax2ax term.

The “difference of two squares” pattern is worth memorising too: the middle terms cancel.

(x+a)2=x2+2ax+a2(x + a)^{2} = x^{2} + 2ax + a^{2}
Square the bracket — do not forget the middle term.
(x+a)(x−a)=x2−a2(x + a)(x - a) = x^{2} - a^{2}
Difference of two squares — the xx terms cancel.
Example 1Expand (x+5)2(x + 5)^{2}
1(x+5)2=(x+5)(x+5)(x+5)^2 = (x+5)(x+5).
2Middle term: 2×5×x=10x2 \times 5 \times x = 10x.
3Last: 52=255^2 = 25.
Answerx2+10x+25x^{2} + 10x + 25

Tip — (x+5)2(x+5)^2 is NOT x2+25x^2 + 25. The cross terms give the 10x10x in the middle.

4

Expanding three brackets

With three brackets, expand two\textbf{two} of them first, then multiply the result by the third bracket. Keep your work neat and collect like terms only at the very end.

Example 1Expand (x+2)(x−1)(x+3)(x + 2)(x - 1)(x + 3)
1Expand the first two: (x+2)(x−1)=x2+x−2(x+2)(x-1) = x^{2} + x - 2.
2Now multiply by (x+3)(x + 3), term by term.
3x2(x+3)=x3+3x2x^{2}(x+3) = x^{3} + 3x^{2};   x(x+3)=x2+3x\;x(x+3) = x^{2} + 3x;   −2(x+3)=−2x−6\;-2(x+3) = -2x - 6.
4Collect like terms: x3+(3+1)x2+(3−2)x−6x^{3} + (3+1)x^{2} + (3-2)x - 6.
Answerx3+4x2+x−6x^{3} + 4x^{2} + x - 6

Tip — It does not matter which two brackets you expand first — pick the pair that looks easiest.

Formula recap

a(b+c)=ab+aca(b + c) = ab + ac
Single bracket — multiply through.
(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^{2} + (a+b)x + ab
Two brackets — FOIL / grid.
(x+a)2=x2+2ax+a2(x + a)^{2} = x^{2} + 2ax + a^{2}
Squaring a bracket.
(x+a)(x−a)=x2−a2(x + a)(x - a) = x^{2} - a^{2}
Difference of two squares.

Common mistakes to avoid

Only multiplying the first term inside a bracket, e.g. 3(x + 2) = 3x + 2.
Multiply every term: 3(x + 2) = 3x + 6.
Writing (x + 5)² = x² + 25.
Expand fully: (x + 5)² = x² + 10x + 25.
Mishandling a negative outside a bracket: −(x − 4) = −x − 4.
Flip every sign inside: −(x − 4) = −x + 4.

Key takeaways

  • Every term inside the bracket is multiplied by every term outside.
  • Two brackets give four products — remember to combine the two middle terms.
  • Squaring a bracket keeps a middle term; difference of two squares loses it.
  • For three brackets, expand a pair first, then multiply by the third.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.